**1.2 Development of early math knowledge**

From the time they are born, children are intrinsically mathematical [20], naturally engaging in mathematical ways of thinking in the areas of number, geometry, measurement, early algebraic reasoning, and data analysis [21, 22]. Developing early mathematical competencies is a complex process that begins well before children enter kindergarten [22] and children need adult support to build and extend their math knowledge [23]. Early childhood teachers and parents1 alike typically recognize verbal counting as an important mathematical skill that is related to later school success [24], but it is mastery of the specific counting principles that is critical to laying the foundation for broader mathematical thinking. For instance, when counting a set of objects, children need to (a) tag one object with one count word without skipping or double counting (one-to-one correspondence), (b) be able to recite number words in the correct order, and (c) know that the last word reached when counting a set represents the whole set (cardinal principle) [25] and children need opportunities to engage in challenging counting activities to develop their understanding of the principles that underlie meaningful counting [26]. Although these principles may seem intuitive to adults, the cardinal principle is a major milestone for preschoolers that leads to new numerical competencies [27, 28] but developing that skill often requires support from both parents and teachers. Given the long-term positive impact that

<sup>1</sup> For the sake of brevity, we sometimes use the word *parent* to refer to children's primary caregivers, but we recognize families come in many configurations, and the primary caregivers may be grandparents, aunts, uncles, older siblings, other family members, or guardians.

early math knowledge has on children's future academic success [1], children need and deserve to build their knowledge of math concepts in their early years.

#### **1.3 Math talk to support children's learning**

Unfortunately, differences are evident in preschoolers' understanding of number and early math concepts [6], which has a cascading effect, as these differences in children's math knowledge predict their math achievement through the elementary school years [12, 29]. A potential explanation for these differences in children's math knowledge is the variability that children experience in their learning environments both at home [30–32] and in preschool classrooms [33]. To build and extend early foundational math skills, children need lots of opportunities to engage in mathematical thinking, in playful ways, that is supported by contingent discussion and "math talk" that allows time for children to think [34]. This is critical as children's mathematical language emerges early, and family engagement in math talk during the early years supports growth in their math skills [35]. Indeed, accumulating evidence suggests that the amount of math talk children experience at home [29] and in preschool [36] and the content of the math talk [37] are correlated with later math skills in elementary school. For instance, young children's number knowledge is related to the amount of family math talk about numbers, starting when children are toddlers [29, 38]. Math talk about key early math concepts such as cardinality [39], spatial reasoning [37, 40, 41], and talking to preschoolers about advanced number concepts [42] have all been shown to predict children's advanced number skills.

#### **1.4 Families and teachers need support in early math**

Children also learn more math when they experience more math interactions at home [30, 43, 44] and in the classroom [45, 46]. Yet many families, including highly educated ones, report that they do not feel confident in their ability to support their child's math learning [47]. A growing body of evidence suggests that when parents interact around mathematics and provide their children with more mathematicsrelated activities and talk, children have higher mathematics outcomes regardless of their family's level of income or education [2, 29, 30, 48, 49]. However, families often do not have access to the types of early math experiences that are developmentally appropriate for preschool children [30, 50] and lack access to the network of supports, resources, and knowledge that are necessary to foster early math development effectively [2]. Importantly, families have highlighted their child's teacher as a key resource that could provide them with more information on what they can do to support their child's math learning [47]. Teachers could provide that bridge for families, which is critically important, as parent-child activities that go beyond counting, such as comparing amounts of items and adding or subtracting objects, can help children acquire more advanced mathematics knowledge [51].

However, early childhood teachers also find it challenging to translate developmentally appropriate teaching strategies into challenging math activities [52–54]. Even in the classroom, preschool teachers often limit their instruction to basic aspects of numbers and counting, and perhaps shapes [55], and spend more time on math concepts that may not be sufficiently challenging [56]. In fact, relatively little time is spent on math learning activities in preschool classrooms at all [33] even though preschoolers who experience more math-related interactions in the classroom and are exposed to more math activities, have higher math achievement [33, 57]. Indeed,

variations in preschool instruction [58] and kindergarten instruction [59] are related to children's math learning. In particular, the percentage of time teachers actively engage children in math, build upon their mathematical ideas, and facilitate children's responses predict gains in their math knowledge [60]. This underscores the need for teachers to understand children's mathematical thinking and use this knowledge not only to guide their teaching [61] but also to support families doing math at home.

### **2. Young Mathematicians**

To enhance opportunities for all children, regardless of background, children need to be provided with learning opportunities that meet their diverse needs [62], and teachers and families need specific tasks to help them see and understand children's mathematical thinking, along with the support and resources to use this information. To address this need, we created the Young Mathematicians (YM) program at EDC, which has partnered with families and early childhood programs to support mathematics learning across home and school environments. YM is grounded in Bronfenbrenner's (1986) Ecological Systems theory [63, 64], which posits that children simultaneously grow and develop within different ecosystems, from the most intimate family and home ecological system, moving outward to the larger school system, and then to the most expansive syst'em: society and culture. Each system inevitably interacts with and influences each other in every aspect of the child's life. Young Mathematicians aim to capitalize on the interconnectedness of children's environments, infusing each level of the ecosystem with positive attitudes toward mathematics and opportunities for children to engage with high-quality early mathematics practices. For the past ten years, we have been working with teachers and families from Head Start2 programs in richly diverse communities with large populations of students of color, linguistically minoritized students, and students living in poverty. Together with these communities, we developed over 55 mathematics games for young children to play at home, at school, or during family play and learning events.

The YM family engagement approach to supporting family math is built on empirical evidence that suggests four important parameters are key for family math interventions to successfully support children's mathematics learning. The first is that a particularly powerful strategy for promoting children's school readiness skills (regardless of content strand), is for schools to provide families with information about the kinds of activities they can do at home that complement children's schoolbased learning [57, 65]. The second is that family math interventions need to provide parents with concrete examples of the mathematics that preschoolers can learn through these daily activities and should illustrate the similarities between early mathematics and language [66]. The third parameter is that families who have a better understanding of early mathematical development may implement more mathematics activities, and therefore family math interventions should focus on developing families' knowledge of young children's mathematical thinking [2]. Finally, early

<sup>2</sup> The federal government funds Head Start programs through the U.S. Department of Health and Human Services, Administration for Children and Families. Eligible participants include 3–5 year old children whose families meet the HHS Poverty Guidelines. Based on the 2023 poverty guidelines [https://aspe.hhs. gov/topics/poverty-economic-mobility/poverty-guidelines], a family of 4 will be eligible for Head Start if their income is at or below \$30,000.

#### *It All Adds Up: Connecting Home and School through Family Math DOI: http://dx.doi.org/10.5772/intechopen.112714*

mathematics interventions should take adults' attitudes toward mathematics into consideration as math anxiety can dampen children's mathematics outcomes [67].

Building on these principles, YM uses games and problem-solving activities (math minibooks) to support young children's foundational mathematics development. YM helps the adults in children's lives to overcome negative feelings about math by offering ideas for how mathematics can be infused into their lives in easy, fun, and playful ways. The materials also provide resources that families can use to expose their children to mathematics-related activities that promote number talk through games, activities, minibooks, and text messages with everyday math ideas. Notably, YM's family engagement approach to family math has helped to underscore for educators the importance of engaging families to support children's learning. Teachers have worked hard to discover new ways of engaging families in early math learning opportunities and in the words of one Head Start teacher who participated in YM:

*We know children's learning begins in the home, families are fundamental in shaping children's interest and skills in math. So, in my opinion we can give families ingredients, and motivation to support their young children's mathematical development effectively. Families can also support children's math development by providing environments that are rich in learning. Families can teach children to see and name small quantities, count, add, subtract, and point out shapes.*

#### **2.1 Why games?**

In order to meet the needs of both families and teachers, we took the approach of designing and testing math games that could be played at home and at school. Math games provide a "hook" for teachers and families—playing a game is more interactive, fun, and developmentally appropriate than, for example, pulling out a worksheet. Play and other informal activities are important contexts in which children develop interests in mathematics, develop their skills, and extend conceptual understanding [45]. Adults can enhance children's exploration and learning through guided play co-playing with children, asking open-ended questions, and exploring materials with them [68]. Math games can therefore promote adult–child co-play with mathematical ideas and foster more math talk in families and in the classroom [68, 69]. When adults and children interact around board games, mazes, and connect-the-dots activities, this helps to promote children's mathematics learning [10, 30, 51]. Games also provide playful learning opportunities that are fun, yet challenging [70] balancing difficulty and skill level, thus fostering motivation and engagement among young children [71]. Additionally, gameplay can promote self-regulation skills through following rules and taking turns, while also offering opportunities for children to practice their skills in communication, empathy, and conflict resolution [71].

For parents, particularly those who may be intimidated by the prospect of "doing math" or simply are not familiar with early math learning and development [47], the game context is also more approachable, as many families welcome incorporating games as part of their family routines and see games as a natural way to play and interact together. The game context can also support adults to engage in mathematics in playful ways, as many adults have had negative experiences with mathematics and may feel some math anxiety that leads them to avoid math. These feelings about math may be barriers to mathematics engagement in the home and may even negatively affect their children's mathematics achievement [72, 73]. Math games can provide families with specific concrete examples of the type of mathematics activities that are developmentally appropriate and fun, which is something that parents say that they want [30, 74] suggesting that a potentially powerful intervention for engaging both teachers and families in mathematics is for children to participate in playful, engaging, and developmentally appropriate mathematics games at home and at school.

Indeed, in our research, we found that adding a family math component to a game-based classroom intervention resulted in positive impacts on preschoolers' mathematics knowledge and was an effective low-threshold intervention that helped to foster early math competencies [57]. For example, some of our games include games with cards and dice. Dice games encourage children to systematically repeat simple counting and adding procedures [75] and card games provide information about number symbols and number words [76] and magnitude comparison [77, 78]. Engaging with children in game-based learning also provides opportunities for adults to observe children's choices and strategies and then provide children with feedback about specific mathematics concepts. The structure of games also helps to support adult caregivers in co-playing while implementing developmentally appropriate instructional strategies. Games are also a fun and developmentally appropriate alternative to the worksheets or flashcards that many caregivers may turn to when they are unsure how to best support children's math learning.

#### **2.2 YM math games**

Together with the Head Start teachers, families, family engagement specialists, librarians, and other educators and partners over the past ten years, we developed over 55 mathematics games for young children to play at home, at school, or during family play and learn events. Resources and instructions in English, Spanish, and Portuguese, including videos and written documentation, are freely available on www.ym.edc.org. The written directions and videos explain how to engage children in the math games and introduce the mathematics concepts that are addressed by each game. The directions encourage players to adapt the games and modify them to their liking. Below we describe two of the games that specifically focus on developing children's knowledge of number.

#### *2.2.1 Numbers, Numbers, 1, 2, 3*

 A favorite game for families and teachers to play is *Numbers, Numbers, 1, 2, 3* because all you need to play with are the fingers on your hands. In this game, children practice counting, knowing how many in all (cardinality), seeing how many immediately (subitizing), and composing and decomposing numbers (see Video 1 in Video Materials). The adult starts by holding their hands behind their back and chanting, "Numbers, numbers, one, two, three, how many fingers do you see?" The adult might show three fingers (one on one hand and two on the other hand). Children might know right away that there are three fingers or they might need to count the fingers—one, two, three. Or they might start counting at "1" but then start over again on the second hand counting "1, 2". The adult can then help them count across both hands "1, 2, 3." Once children figure out how many, you can play again. Children catch onto this game very quickly and typically want to take over being the one showing the fingers on their hands. As children gain more practice, adults can challenge them with numbers up to ten. And later, they can borrow a friend's hands and make numbers up to 20. For a video of how to play the game and an overview of the math for Numbers, Numbers, 1,2,3 go to: https://go.edc.org/Numbers123 (see also Video 1 in Video Materials) [79].

*It All Adds Up: Connecting Home and School through Family Math DOI: http://dx.doi.org/10.5772/intechopen.112714*

#### *2.2.2 Roll Two: a game with cards and dice*

The game *Roll Two* is a developmentally appropriate variation on the game *Shut the Box,* which is rumored to have origins dating back hundreds of years as a popular pub game in England. Our version is appropriate for preschoolers and families and uses two regular six-sided dice and 13 cards printed with numerals 0–12 with the corresponding dice pattern underneath (see **Figure 1**).

Children start by putting a set of cards in order from 1 to 12 (a more challenging variation is 0 to 12, less challenging variations are 1 to 6 or 1 to 3). Adults can learn what children know by watching them arrange the cards: *Do they put them in order from left to right? Do they recognize the numerals and know the order they go in? Do they count the dots on the cards to check which has more and which has fewer? Do they immediately leave a space between four and six to place the five card*. By simply setting up the game, adults learn what children know about numeral recognition, subitizing, and creating a number path. Then, children roll two dice and can turn over cards that match either: one of the rolled numbers; both rolled numbers; or the sum of the two rolled numbers (see **Figure 2**). The goal is to turn over all of the cards. Children can play alone or play against another player to see who can turn over all their cards first. (See also Video 2 in Video Materials for an overview of the math and instructions on how to play the game Roll 2 or go to: https://go.edc.org/Roll2) [80].

While this seems like a simple game, there is a lot of math and a lot of strategy involved. For example, say a child rolls a 3 and a 2. First, they are practicing recognizing dots on the die (subitizing) and connecting that quantity of dots to the numerals 3 and 2 on the cards. They are also practicing combining (or adding) 3 and 2 to make 5. Children may have to count 1, 2, 3, 4, 5 one-by-one to know that these together make 5 or they may be able to recognize 3 and 2 immediately on the dice and be able to count on from

**Figure 1.** *Materials for Roll Two 1–12.*


#### **Figure 2.** *Roll Two directions.*

3—saying 3, 4, 5. With practice, children will be able to quickly add 3 and 2 in their heads and know 5. When playing this game, there are natural opportunities for adults to support children's learning such as encouraging them to use using the counting on strategy. As children gain more practice, they can add a rule that they are also allowed to subtract their roll—if they roll doubles, such as a 4 and a 4—they can subtract 4 from 4 and turn over their 0 card. When introducing subtraction in this game, we always like to add the caveat that children might venture into negative numbers—and that's okay! It can be tempting for adults to say that you cannot take away a bigger number from a smaller number—that you cannot take away 5 from 2—but, of course, you can. It is negative 3. In our materials, we encourage adults not to try to explain negative numbers to preschoolers but also not give children the misconception that it is not possible to subtract a bigger number from a smaller number. We suggest that adults acknowledge that you can take away 5 from 2 and that it will give you a negative number—negative numbers are very interesting and something children will learn about when they are older.
